20191002

kernel

• Cat and Dog problem

• A simple geometric solution

• A more general solution

Dot product \(\vec a\vec b=a_xb_x+a_yb_y=|\vec a||\vec b|\cos(\theta)\)

Exercise:

\(g(x)=<C_+-C_-,X-C>=<C_+,X>-<C_-,X>-<C_+,C>+<C_-,C>\);

\(<C_+,X>=<\frac1{n_{+}}\sum\limits_{l\in I_+}^nx_i,x>\);

\(<C_-,X>=<\frac1{n_{-}}\sum\limits_{l\in I_-}^nx_i,x>\);

\(<C_+,C>=<C_+,\frac12C_+>+<C_+,\frac12C_->=\frac1{2n_{+}^2}\sum\limits_{(i,j)\in I_{+}}<x_i,x_j>+\frac12<C_+,C_->\)

\(<C_-,C>=<C_-,\frac12C_+>+<C_-,\frac12C_->=\frac12<C_+,C_->+\frac1{2n_{-}^2}\sum\limits_{(i,j)\in I_{-}}<x_i,x_j>\)

\(g(x)=\sum_{l=1}^n\alpha_i<x_i,x>+b\),

\(b=\frac12\left[\frac1{n_{-}^2}\sum\limits_{(i,j)\in I_{-}}<x_i,x_j>-\frac1{n_{+}^2}\sum\limits_{(i,j)\in I_{+}}<x_i,x_j>\right]\)

\(\alpha_i=\begin{cases}\frac1{n_{+}}&y_i=+1\\-\frac1{n_{-}}&y_i=-1\end{cases}\)

• Import the iris data

• Define the train and test sets

• Define a kernel function and classifier

• define a 3D grid

• Plot the test and 3D grid

• Another kernel function and another way of computing the classifier

• Define a grid

• Plot the test and grid

• Evaluate the classifier

## [1] 0

Error rate = 0%

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